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Every Lie algebra has a unique single corresponding simply connected Lie group by 
Ciro Santilli 40 Updated 2025-07-16
Ciro Santilli 40 Updated 2025-07-16This simply connected is called the universal covering group.
The orthogonal group has 2 connected components:
- one with determinant +1, which is itself a subgroup known as the special orthogonal group. These are pure rotations without a reflection.
- the other with determinant -1. This is not a subgroup as it does not contain the origin. It represents rotations with a reflection.
It is instructive to visualize how the looks like in :
- you take the first basis vector and move it to any other. You have therefore two angular parameters.
- you take the second one, and move it to be orthogonal to the first new vector. (you can choose a circle around the first new vector, and so you have another angular parameter.
- at last, for the last one, there are only two choices that are orthogonal to both previous ones, one in each direction. It is this directio, relative to the others, that determines the "has a reflection or not" thing
As a result it is isomorphic to the direct product of the special orthogonal group by the cyclic group of order 2:
A low dimensional example:because you can only do two things: to flip or not to flip the line around zero.
Note that having the determinant plus or minus 1 is not a definition: there are non-orthogonal groups with determinant plus or minus 1. This is just a property. E.g.:has determinant 1, but:so is not orthogonal.
Because the people who are crazy enough to think they can change the world are the ones who do.
All action scenes are useless crap, but the premise with Ciro's precious simulation hypothesis subject, related physics and the illusion of life.
It is a shame that the key premise of using human bodies to produce energy is completely and impossibly stupid. You would obviously get more energy by directing burning the food you feed into humans.
If the film had been made later, maybe the much more plausible concept of AI alignment would would have been used instead. What a shame.
Blue Pill or Red Pill scene from The Matrix (1999)
. Source. - www.youtube.com/watch?v=aau8qa3xgFs Neo takes the blue pill mashup with Office Space with deepfake faces
This film is an abomination and disgrace to the bilogy.
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